Electronic band structure of optimal superconductors: From cuprates to ferropnictides...
Transcript of Electronic band structure of optimal superconductors: From cuprates to ferropnictides...
Electronic band structure of optimal superconductors: From cuprates to ferropnictidesand back again (Review Article)
A. A. Kordyuka)
Kyiv Academic University, Kyiv 03142, Ukraine and Institute of Metal Physics, National Academy of Sciencesof Ukraine, Kyiv 03142, Ukraine(Submitted February 14, 2018)
Fiz. Nizk. Temp. 44, 623–634 (June 2018)
While the beginning decade of the high-Tc cuprates era passed under domination of local theories,
Abrikosov was one of the few who took seriously the electronic band structure of cuprates, stress-
ing the importance of an extended Van Hove singularity near the Fermi level. These ideas have
not been widely accepted that time mainly because of a lack of experimental evidence for correla-
tion between saddle point position and superconductivity. In this short contribution, based on the
detailed comparison of the electronic band structures of different families of cuprates and iron-
based superconductors I argue that a general mechanism of the Tc enhancement in all known high-
Tc superconductors is likely related with the proximity of certain Van Hove singularities to the
Fermi level. While this mechanism remains to be fully understood, one may conclude that it is not
related with the electron density of states but likely with some kind of resonances caused by a
proximity of the Fermi surface to topological Lifshitz transition. One may also notice that the elec-
tronic correlations often shift the electronic bands to optimal for superconductivity positions.
Published by AIP Publishing. https://doi.org/10.1063/1.5037550
1. Introduction
In spite of the fashion for the “local language”1–3 in
application to physics of high-Tc cuprates soon after their dis-
covery, some researchers were keeping to believe that it is the
electronic band structure of cuprates that conceals a key to
understand them.4–9 Especially many efforts had been made
to study possible consequences of close vicinity of the “saddle
point” type Van Hove singularity (VHS)10 to the Fermi level
(see Fig. 1), as it had been revealed by the band structure cal-
culations11 and earlier photo-emission experiments in a num-
ber of cuprates.12–14 Abrikosov, who advocated “common
sense against fashion” in this respect, was fascinated by rich
physics that comes from an “extended” saddle-point and had
derived several formulas to describe it analytically12,15–17
suggesting finally his “theory of high-Tc superconducting cup-
rates based on experimental evidence.”18–20
The role of saddle point VHS has been discussed in two
types of scenarios. The “direct” scenarios relate the Tc enhance-
ment with VHS related peak in the density of states (DOS),7,9,21
which for the “extended” singularity8,12 leads to the stronger
than logarithmic divergence (“a power law divergence”) in
DOS.15,19 In the “indirect” scenarios, the superconductivity is
enhanced by competing instabilities.4–6,22,23 Moreoover, it has
been found that strong correlation effects pin this VHS close to
the Fermi level.24–29 Other aspects of the saddle point VHS,
like the dynamic VHS-Jahn–Teller effect, the pseudogap and
striped phases, are discussed in detail in another review by
Markiewicz.30
The discussed singularity in DOS was later shown to be
rather weak to account for high Tc’s, especially when finite
temperature and impurity scattering are taken into account.31,32
Applicability of the models with competing instabilities is
more difficult to estimate but the overall frustration about
them have been arisen mainly because of a lack of experimen-
tal evidence for correlation between the position of the saddle
point and superconductivity: a number of cuprates with VHS
close to the Fermi level show rather low Tc’s.11,13,14 In addi-
tion, while the saddle point stays close to the Fermi level for
the hole-doped cuprates, it goes deeper with hole underdoping
and should continue to sink further with the electron doping.
So, any scenarios of Tc enhancement related to the saddle point
VHS cannot universally explain the both sides of the electronic
phase diagram.
Here, based on overview of a number of photoemission
data, I argue that indeed there is a robust correlation between
superconducting critical temperature and a proximity of cer-
tain Van Hove singularities to the Fermi level in all known
high-Tc superconductors including the iron-based supercon-
ductors (Fe-SC) and high-Tc cuprates (CuSC) on the both
sides of the phase diagram. Interestingly, this VHS is usually
not a saddle point but an edge (top or bottom) of certain
bands which, in the vicinity to topological Lifshitz transi-
tion33 plays, most likely, a role of a “resonant amplifier” of
superconductivity in a multi-band system. While we are look-
ing for microscopic understanding of this correlation, it can
be used to search for new high-temperature superconductors
with higher Tc’s.
2. Electronic band structure
2.1. Cuprates
As it has been mentioned, when Abrikosov was explor-
ing the consequences of the extended saddle point in the elec-
tronic band structure of the cuprates, most of the researchers
did not believe that the concepts of the one-particle electronic
structure or the Fermi liquid are applicable to the cuprates at
all. However, already the first angle resolved photoemission
experiments on Bi2Sr2CaCu2O8þx (BSCCO or Bi-2212),34,35
YBa2Cu3O7–d (YBCO),36 and Nd2–xCexCuO4 (Ref. 37)
had revealed the dispersion and the Fermi surface very
similar to those obtained by conventional density-functional
1063-777X/2018/44(6)/10/$32.00 Published by AIP Publishing.477
LOW TEMPERATURE PHYSICS VOLUME 44, NUMBER 6 JUNE 2018
band-structure calculations. The essential development of
angle resolved photoemission spectroscopy (ARPES) during
the next decade has allowed to shed much more light on this
issue revealing the details of the band structure and quasipar-
ticle spectrum of cuprates.38,39
In particular, it has been shown that the hole-doped cup-
rates have a large Fermi surface (FS) in the range of doping
when they are superconducting.40–43 This FS satisfies the
Luttinger theorem, i.e., its volume corresponds to the number
of the conduction electrons per unit cell and is proportional to
(1� x), where x is the hole concentration. Then,43–47 most of
ARPES-groups began to observe the splitting of the conduc-
tion band in the bilayer cuprates into the sheets corresponding
to the bonding and anti-bonding orbitals (see Fig. 2) that con-
tradicted the idea of spatial confinement of electrons in sepa-
rate layers.1,48 Note that most of ARPES results for cuprates
have been obtained on Bi-2212, since the bulk properties of
Y-123 are much more difficult to study because of overdoped
non-superconducting topmost layer.49,50
Based on the FS geometry and low-energy electron dis-
persions one may derive the hopping integrals describing the
conduction band51 and, for two-CuO2-layer Bi-2212 we have
derived that the onset of the superconducting region in the
phase diagram, in the direction of reducing the hole concen-
tration, starts with the Lifshitz topological transition for the
antibonding Fermi surface52 [Fig. 2(g)]. While the same holds
for a single FS of single layer Bi-2201,53 a careful study of
Bi-2212 of different doping levels54 has shown that the
Lifshitz transition for the antibonding band appears a bit later,
at between 0.22 and 0.23 holes per Cu atom (Tc¼ 55 K), that
is similar to earlier result on La2–xSrxCuO4.55
Another conclusion that has been derived from the numer-
ous ARPES experiments is that the whole spectra of the super-
conducting cuprates (except may be the pseudogap effect)
can be described by the quasiparticle spectral function,52,56
A/ Im (G), G �1¼G�10 �R, in which the bare Green’s func-
tion G0 ¼ 1=ð ek � xþ i0Þ with the bare electron dispersion
ek are defined by the interaction of the electrons with periodic
crystal lattice and the quasiparticle self-energy R encapsulates
the interaction of electron with other electrons and other
degrees of freedom, like in normal metals.57 Yet the striking
difference of ARPES spectra of the cuprates from the spectra
of normal metals is in strong scattering that (1) does not stop at
the Debye energy [so, the dispersion is hard to follow below
�0.3 eV (Refs. 58 and 59)] and (2) is strongly momentum
dependent, leading to the “nodal-antinodal dichotomy”: around
(p,0) point of the Brillouin zone (the “antinodal” region)
the renormalization is highly increasing below Tc, while along
the nodal direction its temperature dependence is rather
weak.52,60,61
The analysis of the self-energy dependence on energy,
momentum,61,62 temperature and doping level58,63,64 has indi-
cated that the main channel of one-electron excitation scatter-
ing is related with the spin-fluctuations. The direct comparison
of the ARPES and inelastic neutron scattering spectra (Fig. 3)
Fig. 2. Electronic band structure of an overdoped Bi-2212 derived from ARPES experiments: a fragment of the Fermi surface (arrows indicate the nodal FS
crossings with well resolved bilayer splitting) (a);47 the ARPES spectrum represents the antinodal crossing of the saddle point (b);60 the bare band structure
(c), FS contours (d), dispersions (e) and DOS (f) derived from experimental data;51 the position of the saddle point of the antibonding band derived from the
data (EDC maximum, black circles) and bare band positions (eA,blue squares) in meV are shown together with Tc (grey solid line) in K as function of hole con-
centration (g).52
Fig. 1. The “extended saddle point” type Van Hove singularity derived from
photoemission data from YBa2Cu4O8: The momentum axes cover an inter-
val of 1 A�1 centered on (p,0) point of the Brillouin zone, and the binding
energy axis spans 0–60 meV.12
478 Low Temp. Phys. 44 (6), June 2018 A. A. Kordyuk
has proved this idea,65 naturally resolving the nodal-antinodal
dichotomy: the self-energy of the nodal quasiparticles is
defined by scattering by high-energy branches of the spin-
fluctuation spectrum while the antinodal self-energy is formed
by the scattering by the magnetic resonance66 formed below Tc.
We can consider R as a generalized cross-correlation56 of one-
particle spectrum presented by the Green’s function and the
two-particle spectrum of spin-fluctuations:P¼ �U
2v�G,
where �U is taken for a spin-electron coupling constant. Also, it
has been shown67,68 that the spin-fluctuation spectrum itself is
formed by itinerant electrons: v¼G ? G. So, one can write an
extended Dyson equation for Cu-SC (see Ref. 56 for details)
G�1 ¼ G�10 � �U
2G � G � G: (1)
Therefore, one could conclude that “a conservative
view” of Friedel8 in late 80’s, that treating the on-site elec-
tron interactions as a perturbation to a band scheme is suffi-
cient to describe the physical properties of superconducting
cuprates, is largely supported by later ARPES experiments.
The issue of the pseudogap in cuprates is rather complicated
since evidently encapsulates a number of mechanisms,69 some
of which, like charge or spin density waves can be described by
an effect of the new order, but still there is a place for localiza-
tion effects. I will turn back the pseudogap issue in Sec. 3.2.
To summarize, the electronic band structure (ES) of
cuprates defines the spin-fluctuation spectrum (SF) and the
electronic ordering, which likely forms the pseudogap (PG)
state. An interplay of ES with SF leads to superconductivity
(SC) that competes with PG. So, one can write one more for-
mula for Cu-SC:
(2)
The pairing by the spin-fluctuations can lead to high Tc in
some theories70–72 or cannot in point of view of others.73,74
2.2. Iron-based superconductors
The band structure of the iron-based superconductors
(FeSC) is much more complex than of Cu-SC and consists
usually of five conduction bands crossing the Fermi level75,76
(see Fig. 4). It is well captured by DFT calculations77,78 but
do not take it too literally. The calculated Fermi surface is
usually bad starting point for theory, since even topology of
the Fermi surface is very sensitive to slight shifts of the bands
in respect to EF and to each other and often differs in experi-
ment and calculations.79,80
The band structures of Fe-SC seen in the ARPES experi-
ments differ from the calculated ones mainly in two ways:
a strong renormalization [3 times in avarage80,81 but band-
dependent82,83 and peaked at about 0.5 eV (Ref. 84)] and a
momentum-dependent shift80,85–87 (the “red-blue shift”88,89).
The bands forming the Fermi surfaces of Fe-SC have dis-
tinct orbital characters mainly of three types: Fe 3dxy, 3dxz, 3dyz
[Figs. 4(b)–4(d)]. Moreover, it is the dxz=dyz bands that carry
the largest superconducting gap90–93 and are therefore the most
Fig. 3. “Fingerprints” of the spin-fluctuations in ARPES spectra of cuprates. (a) The Fermi surface of YBCO in the 1st Brillouin zone derived from ARPES
data represents the fermionic Greens function. (b) The spin excitations spectrum resulting from numerical fits to the inelastic neutron scattering data. (c)
Comparison of experimental (upper row) and theoretical (lower row) fermionic spectra.56,65
Fig. 4. Electronic structure of iron-based superconductors (Fe-SC): (a) Fermi surface (FS) maps measured by ARPES for an optimally doped Ba1–xKxFe2As2
(BKFA);79 (b) the scketch of FS derived from comparison of experimental spectra to band structure calculations with indication (by color) of the orbital char-
acted of corresponding bands; the comparison of the calculated (c, and dotted lines on d) and experimental (d) band structure along the UX direction of the
Brillouin zone.75,76
Low Temp. Phys. 44 (6), June 2018 A. A. Kordyuk 479
important for superconductivity in Fe-SC. This simplifies the
situation a bit, but, on the other side, the AF ordered phase and
preceeding nematic transition94 essentially complicates the
electronic band structure of the “normal” state from which the
superconductivity occurs.
From the theory side, the question what drives both the
supercoducting pairing and the nematic ordering remains
open. The phonons alone, despite some “fingerprints” they left
in the ARPES spectra,81 are not considered seriously.94,95
Even the state is called “nematic” rather than “anisotropic” to
stress the electronic origin of the instability, and there is a lot
of experimental evidences for this,94 but one prefers to speak
about interplay of phonons, charge/orbital fluctuations, and
spin fluctuations.94,96–99 Although it is agreed that phonons
and charge/orbital fluctuations would favour a sign-preserving
s-wave superconducting order parameter (sþþ) whereas spin-
fluctuations favour a sign-changing s-wave (sþ�) supercon-
ductivity,94,95,100 any agreement on why Tc’s are so high is
absent so far and there is no confirmed prediction for new
high-temperature superconductors.
3. Lifshitz transition
3.1. Iron-based superconductors
From the experiment side, the complexity of the band
structure of Fe-SC seems to play a positive role in the strug-
gle for understanding the pairing mechanism because the
multiple electronic bands and the resulting complex fermiol-
ogy offer exceptionally rich playground for establishing use-
ful empirical correlations. In particular, there is an empirical
correlation between the electronic structure and Tc:: maximal
Tc (optimally doped superconductors) is observed when a
proximity of the electronic structure to topological Lifshitz
transition33 takes place.75,76 Interestingly, this Lifshitz tran-
sition (LT) is related with VHS wich is usually not a saddle
point but an edge (top or bottom) of certain bands, namely
dxz=dyz bands, as shown in Fig. 5.
This LT-Tc correaltion is observed for all known optimally
doped Fe-SC except, may be, some FeSe-based com-
pounds.88,101 Indeed, the extremely small Fermi surface sheets
of dxz=dyz orbital origin are observed for the optimally hole-
doped Ba1–xKxFe2As2 (BKFA),79,91,102 Ba1–xNaxFe2As2
(BNFA),103 and Ca1–xNaxFe2As2 (Ref. 104) as well as for the
optimally electron doped Ba(Fe1–xCox)2As2 (BFCA),76,105 i.e.,
for the both sides of the electron phase diagram for the 122 sys-
tem. The same holds for the stoichiometric (but optimal for
Tc LiFeAs,80,81,92 NaFeAs,106,107 and for AxFe2–ySe2 family (A
stands for alkali metal: K, Rb, Cs, and Tl).108–110 Moreover, the
Tc is increasing with the number of band-edge VHS’s at EF, as
it has been shown comparing (CaFeAs)10Pt3.58As8 (three band-
edge VHS’s at EF, Tc¼ 35 K) to (CaFe0.95Pt0.05As)10Pt3As8
(only one VHS at EF, Tc¼ 15 K).111 Finally, now one can
say that the same is true for the 1111-type compounds
which exhibit the highest Tc up to 55 K. Having the polar surfa-
ces, these compounds are hard to study by ARPES,39,75 but
it has been shown that the bulk electronic structure for
SmFe0.92Co0.08AsO112 and NdFeAsO0.6F0.4 (Tc¼ 38 K)113 is in
the same optimal for superconductivity state, having 2–3 band-
edge VHS’s in close vicinity to the Fermi level.
As for FeSe, its Fermi surfaces look a bit away from the
Lifshitz transition,88 but pure FeSe crystals are not optimal
for superconductivity: their Tc increases from about 9 to 38 K
under pressure114 and by means of intercalation.115 While it
is hard to do ARPES under such a pressure, the results of a
DFT þ DMFT calculations show that the bulk FeSe under
pressure about 9 GPa undergoes a Lifshitz transition.116
The possible mechanism of this correlation will be briefly
discussed in Sec. 3.3, but it would be tempting to use the
observed correlation for a search of new high-temperature
superconductors with higher Tc’s. Similar electronic band
structure for all the Fe-SC’s results in similar DOS76 from
which one can clearly see that this correlation has nothing to
do with DOS enhancement. The bright example is KFe2As2
Fig. 5. The LT-Tc correlation75,76 is illustrated through a projection of the Fermi level crossing the “rigid” electronic band structure of Fe-SC (central panel)
on the charge carrier concentration scale of the phase diagram (right): Tc maxima correspond to proximity of the tops/bottoms of dxz=dyz bands to the Fermi
level; the corresponding “optimal” Fermi surfaces (blue for hole- and red for electron-like sheets) are shown on the left.
480 Low Temp. Phys. 44 (6), June 2018 A. A. Kordyuk
that has much higher DOS at EF than any of optimally doped
Fe-SC’s and Tc about 4 K. On the other hand, the density of
states should be certainly important for superconductivity, so,
looking for Fe-SC’s with higher Tc one should find a com-
pound with several bands crossing the Fermi level one or
more of which are close to the Lifshitz transition but with the
higher density of states from the other bands (similarly to
hole-doped 122). This should be the case for hole overdoped
KFe2As2 or LiFeAs.
3.2. Cuprates
Unexpectedly, the recent progress in understanding the
mechanisms of pseudogap formation in cuprates (see Ref. 69
for review) leads to conclusion that the same LT-Tc correla-
tion takes place also for cuprates (both for the hole- and the
electron-doped ones) in the antiferromagnetic (AF) Brillouin
zone, i.e., assuming that the pseudogap is caused by an
AF-like electronic ordering.
Indeed, it has been shown that while several mechanisms
contribute to the pseudogap phenomenon, a short range or
slightly incommensurate117 AF-like ordering stays mostly
responsible for the pseudogap openning below T*.69 This
ordering is most likely a result of VHS nesting,118 that is the
known mechanism for electronic ordering in “excitonic insu-
lators.”119 Some evidence for incommensurate spin-density
wave (SDW) has been obtained in neutron experiments on
YBCO,120 while in Refs. 118 and 121 it has been shown
that temperature evolution of antinodal ARPES spectrum for
Bi-2201 is mostly consistent with a commensurate (p,p) den-
sity wave order.
This means that the superconductivity in cuprates with
the highest Tc appears in the AF-ordered “normal” state, the
Fermi surfaces of which are shown on the left side of Fig. 6
for the electron (top) and hole (bottom) doped sides of the
phase diagram shown on the right. The most representative
cut of the electronic bands, taken along the AF Brillouin zone
boundary, is shown in center of Fig. 6. The two bands shown
here are the result of the hybridization between original CuO
band and its replica folded into the AF Brillouin zone.
One can see that maximal Tc’s are observed when either
higher (red) or lower (blue) band are in close vicinity to
Lifshitz transition for the hole- and electron-doped cuprates,
respectively, that is intriguingly similar to the Fe-SC case
discussed above.
The splitting between these two bands depends of the
mechanism of the “AF-like” ordering, that brings us to the old
discussion on Slater vs Mott insulators.8 Effective doubling of
the unit cell can be described in many ways: as Peierls or spin-
Peierls122 type instability powered by the VHS nesting,119 in
the extended Hubbard model,123 and in the tJ -model124,125 in
which the quasiparticles cannot leave the magnetic sublattice
in which they were created.126 So, the two old but related
questions are arising again: (1) whether it possible to decide
between Slater and Mott scenarios based on ARPES data and
(2) how this mechanism does affect the electronic structure
and, subsequently, the transition to superconducting state.
In my opinion, for the hole-doped cuprates, this question
can be clarified looking for the spectral weight which disap-
pears with pseudogap opening below T* but reappears below
Tc.69 The upper band (in the center panel of Fig. 6) is not
clearly visible in ARPES spectra,118,121 likely because of
short range or incommensurate117 character of the ordering.
One can also remind here the complication that comes from
the “shadow band,”40 later attributed to structural modula-
tions,127,128 and the bilayer splitting,43–47 that further com-
plicates the spectra of the bilayer cuprates. So, in order to
describe the band gap caused by the AF-like ordering one
needs a detailed temperature-dependent ARPES study of
preferably one layer compounds.
The situation is much simpler for the electron-doped cup-
rates.129 Despite the chose between Slater and Mott pictures is
also discussed here,130,131 the ARPES data clearly shows a gap
along the magnetic zone boundary132,133 and the Fermi surface
like in Fig. 6 (left-bottom), confirming the AF doubling of the
unit cell.
Fig. 6. Similar projection of the electronic band structure of high-Tc cuprates, represented by the dispersion along the (p,0)–(0,p) cut of large Brillouin zone
(central panel), on their phase diagram (right):69 Tc maxima correspond to proximity of the tops/bottoms of the antiferomagnetically folded bands to the Fermi
level; the corresponding Fermi surfaces (blue for hole- and red for electron-like) are shown on the left panels for EeF (top) and Eh
F (bottom) Fermi levels.
Low Temp. Phys. 44 (6), June 2018 A. A. Kordyuk 481
To summarize, the AF-like ordering in cuprates puts
them in a row with the Fe-SC’s following the empirical cor-
relation that highlights the importance of topological Lifshitz
transition for high-temperature superconductivity. One may
speculate that the presence of small Fermi surfaces is a gen-
eral mechanism for enhancement of super-conductivity that
is usually observed at the charge/spin density wave phase
boundary in a number of quasi-2D systems, and that it is the
size or geometry of these small Fermi surfaces rather than
an enhancement of the density of states that powers this
mechanism.
3.3. Resonant superconductivity?
The question about possible mechanism of Tc enhance-
ment by a proximity to the Lifshitz transition is evidently
not straightforward since the very definition of this transition
as the “2.5 phase transition” in the terminology of Ehrenfest
indicates that one may expect (for 3D system) to observe sin-
gularity only in between the 2nd and 3rd derivatives of the
thermodynamic potential: z1/2 and z�1=2 singularities, respec-
tively, where z¼EF� eVHS is the energy distance of band-
edge VHS to the Fermi level.33
The effect of Lifshitz transition on electronic properties
of metals has been reviewed in Refs. 134 and 135. In particu-
lar, it was shown that, besides evident appearance of cusps
in DOS and consequently in the heat capacity, magnetic sus-
ceptibility, etc., it leads to a special channel of scattering
with a specific energy-dependent correction in the electron
relaxation time, which is responsible for a giant anomaly in
the thermoelectric power and other kinetic characteristics of
metals. Although the effect on superconductivity has not
been discussed in those reviews, one may expect a similar
effect since often the growth of thermopower is correlated
with the changes in Tc. Moreover, in anisotropic supercon-
ductors, the change in Tc due to the change in the Fermi-
surface topology becomes stronger and nontrivial.136
Both Fe-SC’s and Cu-SC’s are quasi-2D materials with the
step-like singularity in DOS at eVHS instead of ðe� eVHSÞ1=2
for 3D compounds, so, one may expect stronger effects here. In
addition, they are multi-band superconductors, in which these
effects can be enhanced. For example, a “superlinear” enhance-
ment (with number of valleys) of effective coupling constant
has been predicted within the BCS model for multi-valued
semimetals taking into account inter-valley coupling.137
So, one may expect that the microscopic explanation for
the observed LT-Tc correlation may be related with some
LT-powered “resonant amplifier” of superconducting pairing
in a multi-band system. This said, the superconductivity in
FeAs-based compounds has been assigned to a Feshbach res-
onance (also called “shape resonance”138) in the exchange-
like interband pairing.139 This mechanism has been further
developed in140–143 for a number of systems, including the
potassium doped p-Terphenyl144 with Tc up to 123 K.
Going back to the issue of superconductivity in Fe-SC
system, one may subdivide the pairing problem into “glue”
and “amplifier.” Recently,145 we have added an argument
to support the sþ� scenario based on the phase sensitive
Josephson junction experiment. Taking the spin-fluctuations
as a main glue for electron pairs in Fe-SC, one may consider
the shape resonance as the mechanism for an entanglement
of spin and orbital degrees of freedom.
To summarize, one may assume that the proximity of
Fermi surface to topological transition is a universal feature
for Tc enhancement mechanism. While this mechanism
remains to be fully understood, one may notice that the elec-
tronic correlations often shifts the electronic bands to opti-
mal for superconductivity positions. In respect to cuprates,
there were a number of experimental evidences and theoreti-
cal treatments for the effect of pinning of the saddle-point
VHS to the Fermi level. The band-edge VHS, discussed
here, is formed due to AF ordering and tuned by the pseudo-
gap value. So, the role of the PG in high-Tc story is not just
in competition with superconductivity for the phase space,
as suggested by Eq. (2), but also in shifting the upper split
band (for the hole-doped cuprates) to the Lifshitz transition.
Speculating more, this position should be very sensitive to
the new order potential and can be pinned to EF to minimize
the electron kinetic energy.
As for the Fe-SC compounds, it is interesting to note
that the observed “red-blue shift”88,146 can be a consequence
of similar pinning mechanism.
4. “Red-blue shift”
When one compares the band structure of iron-based
superconductors derived from ARPES experiment with the
result of DFT calculations, one can see that it is not “rigid”
but distorted by a momentum-dependent shift that acts simi-
larly in all Fe-SC’s, shifting the bands up and down in
energy: up—in the center of the Brillouin zone and down—
in its corners,76,80,85,86 as shown in Fig. 7). Since such a shift
persists in all the Fe-SC’s and is a sort of natural degree of
freedom for the band structure of a multi-band metal with
the Luttinger-volume conserved, that results in synchronous
change (shrinking, in this case83,87,147) of the hole and elec-
tron Fermi surfaces, it is tempting to give it a special name
and, following Refs. 88 and 89, I call it the “red-blue shift”
here.
Since it is the electron interactions that are missing in
DFT calculations, it is also tempting to ascribe such a shift
to these interactions, for which several models have been
proposed. The Fermi surface shrinking can be a consequence
of the strong particle-hole asymmetry of electronic bands
assuming a dominant interband scattering148 and described
by the self-energy corrections due to the exchange of spin
fluctuations between hole and electron pockets,83,149 or it
can be formulated in terms of s-wave Pomeranchuk order-
ing.99 On the other hand, one can explain it as a decrease of
a band width due to a screening of the nearest neighbor hop-
ping as a result of AF-like ordering88 that, similarly to cup-
rates, can be considered as a consequence of the confinement
of the carriers within the magnetic sublattice.126 One should
note that all these mechanisms will lead to Lifshitz transition
for non-compensated carriers, that requires hole or electron
doping to shift from stoichiometry, multiple bands or both.
If the discussed shift is a result of correlations, one may
expect its enhancement with lowering temperature.88 Such
temperature evolution of the band structure is in agreement
with Hall measurements102 and has been observed recently
by ARPES on FeSe crystals.146 This results, however, is not
482 Low Temp. Phys. 44 (6), June 2018 A. A. Kordyuk
confirmed by other experiments,150–152 so, one may conclude
that the temperature effect is more complex and requires fur-
ther research.
5. Summary
The electronic band structure of cuprates defines both
the spectrum of the spin-fluctuations, which bound electrons
in pairs, and the AF-like electronic ordering, which forms
the pseudogap state. The band structure of the iron-based
superconductors is much more complex than of cuprates.
The pairing can be due to spin-fluctuations, phonons or both,
but why the Tc’s are so high is not clear. Nevertheless, there
is an empirical correlation between electronic structure and
Tc: maximal Tc (optimally doped SC) is observed when
proximity of the ES to topological Lifshitz transition takes
place. This is observed for all Fe-SCs.
Interestingly, the same correlation holds for Cu-SC
(both for hole- and electron-doped ones) in the antiferromag-
netic Brillouin zone, i.e., assuming that the PG is caused by
the AF-like electronic ordering. So, an interplay of the elec-
tronic structure with the spin-fluctuations leads to supercon-
ductivity that, on one hand, competes with the pseudogap
caused by the AF-ordering, but, on the other hand, can be
enhanced by the proximity to Lifshitz transition, also caused
by this ordering.
The idea of this review was to stress ones more that this
correlation is either annoyingly observed by ARPES or pre-
dicted by calculations for all the known high- super-conductors.
This allows to assume that the proximity of Fermi surface to
topological transition is a universal feature for a Tc enhance-
ment mechanism. While we are looking for microscopic under-
standing of this correlation, it can be used to search new
high-temperature superconductors with much higher transition
temperatures.
I acknowledge discussions with A. Abrikosov, L. Alff,
A. Avella, A. Bianconi, B. B€uchner, S. V. Borisenko, V.
Brouet, A. V. Chubukov, T. Dahm, C. Di Castro, I. Eremin,
D. V. Evtushinsky, J. Fink, A. M. Gabovich, M. S. Golden, M.
Grilli, D. S. Inosov, I. N. Karnaukhov, A. L. Kasatkin, T. K.
Kim, M. M. Korshunov, V. M. Krasnov, A. Lichtenstein, V.
M. Loktev, R. Markiewicz, I. V. Morozov, S. G. Ovchinnikov,
E. A. Pashitskii, N. M. Plakida, V. M. Pudalov, Yu. V.
Pustovit, M. V. Sadovskii, D. J. Scalapino, A. V. Semenov, J.
Spałek, T. Valla, A. A. Varlamov, A. N. Vasiliev, A. N.
Yaresko, and V. B. Zabolotnyy. The project was supported by
the Ukrainian-German grant from the MES of Ukraine
(Project No. M/20-2017) and by the Project No. 6250 by
STCU and NAS of Ukraine.
a)Email: [email protected]
1P. W. Anderson, Science 235, 1196 (1987).2E. Dagotto, Rev. Mod. Phys. 66, 763 (1994).3M. Imada, A. Fujimori, and Y. Tokura, Rev. Mod. Phys. 70, 1039 (1998).4J. E. Hirsch and D. J. Scalapino, Phys. Rev. Lett. 56, 2732 (1986).5I. E. Dzialoshinskii, Sov. Phys. JETP 66, 848 (1987).6A. Gorbatsevich, V. Elesin, and Y. Kopaev, Phys. Lett. A 125, 149
(1987).7J. Labb�e and J. Bok, Europhys. Lett. 3, 1225 (1987).8J. Friedel, J. Phys.: Condens. Matter 1, 7757 (1989).9R. Markiewicz and B. Giessen, Physica C 160, 497 (1989).
10L. Van Hove, Phys. Rev. 89, 1189 (1953).11R. Markiewicz, J. Phys. Chem. Solids 58, 1179 (1997).12K. Gofron, J. C. Campuzano, A. A. Abrikosov, M. Lindroos, A. Bansil,
H. Ding, D. Koelling, and B. Dabrowski, Phys. Rev. Lett. 73, 3302
(1994).13D. M. King, Z.-X. Shen, D. S. Dessau, D. S. Marshall, C. H. Park, W. E.
Spicer, J. L. Peng, Z. Y. Li, and R. L. Greene, Phys. Rev. Lett. 73, 3298
(1994).14T. Yokoya, A. Chainani, T. Takahashi, H. Katayama Yoshida, M. Kasai,
and Y. Tokura, Phys. Rev. Lett. 76, 3009 (1996).15A. Abrikosov, J. Campuzano, and K. Gofron, Physica C 214, 73 (1993).16A. Abrikosov, Physica C 222, 191 (1994).17A. A. Abrikosov, Phys. Rev. B 57, 8656 (1998).18A. A. Abrikosov, Int. J. Mod. Phys. B 13, 3405 (1999).19A. Abrikosov, Physica C 341–348, 97 (2000).20A. A. Abrikosov, Physica C 468, 97 (2008).21J.-H. Xu, T. J. Watson-Yang, J. Yu, and A. J. Freeman, Phys. Lett. A
120, 489 (1987).22R. S. Markiewicz, J. Phys.: Condens. Matter 3, 3859 (1991).23A. P. Kampf and A. A. Katanin, Phys. Rev. B 67, 125104 (2003).
Fig. 7. The “red-blue shif” of the experimental band structure (the solid lines in four panels on the left) of the FeSe single crystal in comparison to the calcu-
lated one (the dotted lines) shown by the red and blue arrows88 and similar shifts with lowering temperature for the tops of the dxz and dyz bands in the center
(top right panel) of the Brillouin zone and for the merging point of these bands in its corner (bottom right panel).146
Low Temp. Phys. 44 (6), June 2018 A. A. Kordyuk 483
24R. S. Markiewicz, J. Phys.: Condens. Matter 2, 665 (1990).25D. M. Newns, P. C. Pattnaik, and C. C. Tsuei, Phys. Rev. B 43, 3075
(1991).26Q. Si and G. Kotliar, Phys. Rev. B 48, 13881 (1993).27P. Monthoux and D. Pines, Phys. Rev. B 47, 6069 (1993).28O. K. Andersen, O. Jepsen, A. I. Liechtenstein, and I. I. Mazin, Phys.
Rev. B 49, 4145 (1994).29A. I. Liechtenstein, O. Gunnarsson, O. K. Andersen, and R. M. Martin,
Phys. Rev. B 54, 12505 (1996).30R. S. Markiewicz, Phys. Rev. B 56, 9091 (1997).31E. A. Pashitskii and V. I. Pentegov, Fiz. Nizk., Temp. 32, 596 (2006)
[Low Temp. Phys. 32, 452 (2006)].32N. Plakida, High-Temperature Cuprate Superconductors: Experiment,
Theory, and Applications, Springer Series in Solid-State Sciences
(Springer, 2010).33I. M. Lifshitz, Sov. Phys. JETP 11, 1130 (1960).34T. Takahashi, H. Matsuyama, H. Katayama-Yoshida, Y. Okabe, S.
Hosoya, K. Seki, H. Fujimoto, M. Sato, and H. Inokuchi, Nature 334, 691
(1988).35D. S. Dessau, Z.-X. Shen, D. M. King, D. S. Marshall, L. W. Lombardo,
P. H. Dickinson, A. G. Loeser, J. DiCarlo, C.-H. Park, A. Kapitulnik, and
W. E. Spicer, Phys. Rev. Lett. 71, 2781 (1993).36R. Liu, B. W. Veal, A. P. Paulikas, J. W. Downey, P. J. Kostic, S.
Fleshler, U. Welp, C. G. Olson, X. Wu, A. J. Arko, and J. J. Joyce, Phys.
Rev. B 46, 11056 (1992).37D. M. King, Z.-X. Shen, D. S. Dessau, B. O. Wells, W. E. Spicer, A. J.
Arko, D. S. Marshall, J. DiCarlo, A. G. Loeser, C. H. Park, E. R.
Ratner, J. L. Peng, Z. Y. Li, and R. L. Greene, Phys. Rev. Lett. 70, 3159
(1993).38A. Damascelli, Z. Hussain, and Z.-X. Shen, Rev. Mod. Phys. 75, 473
(2003).39A. A. Kordyuk, Fiz. Nizk. Temp. 40, 375 (2014) [Low Temp. Phys. 40,
286 (2014)].40P. Aebi, J. Osterwalder, P. Schwaller, L. Schlapbach, M. Shimoda, T.
Mochiku, and K. Kadowaki, Phys. Rev. Lett. 72, 2757 (1994).41S. V. Borisenko, M. S. Golden, S. Legner, T. Pichler, C. D€urr, M.
Knupfer, J. Fink, G. Yang, S. Abell, and H. Berger, Phys. Rev. Lett. 84,
4453 (2000).42S. V. Borisenko, A. A. Kordyuk, S. Legner, C. D€urr, M. Knupfer, M. S.
Golden, J. Fink, K. Nenkov, D. Eckert, G. Yang, S. Abell, H. Berger, L.
Forr�o, B. Liang, A. Maljuk, C. T. Lin, and B. Keimer, Phys. Rev. B 64,
094513 (2001).43A. A. Kordyuk, S. V. Borisenko, M. S. Golden, S. Legner, K. A. Nenkov,
M. Knupfer, J. Fink, H. Berger, L. Forr�o, and R. Follath, Phys. Rev. B 66,
014502 (2002).44D. L. Feng, N. P. Armitage, D. H. Lu, A. Damascelli, J. P. Hu, P.
Bogdanov, A. Lanzara, F. Ronning, K. M. Shen, H. Eisaki, C. Kim, Z.-
X. Shen, J.-I. Shimoyama, and K. Kishio, Phys. Rev. Lett. 86, 5550
(2001).45Y.-D. Chuang, A. D. Gromko, A. Fedorov, Y. Aiura, K. Oka, Y. Ando,
H. Eisaki, S. I. Uchida, and D. S. Dessau, Phys. Rev. Lett. 87, 117002
(2001).46A. A. Kordyuk, S. V. Borisenko, T. K. Kim, K. A. Nenkov, M. Knupfer,
J. Fink, M. S. Golden, H. Berger, and R. Follath, Phys. Rev. Lett. 89,
077003 (2002).47A. A. Kordyuk, S. V. Borisenko, A. N. Yaresko, S.-L. Drechsler, H.
Rosner, T. K. Kim, A. Koitzsch, K. A. Nenkov, M. Knupfer, J. Fink, R.
Follath, H. Berger, B. Keimer, S. Ono, and Y. Ando, Phys. Rev. B 70,
214525 (2004).48H. Ding, A. F. Bellman, J. C. Campuzano, M. Randeria, M. R. Norman,
T. Yokoya, T. Takahashi, H. Katayama-Yoshida, T. Mochiku, K.
Kadowaki, G. Jennings, and G. P. Brivio, Phys. Rev. Lett. 76, 1533
(1996).49V. B. Zabolotnyy, S. V. Borisenko, A. A. Kordyuk, J. Geck, D. S. Inosov,
A. Koitzsch, J. Fink, M. Knupfer, B. B€uchner, S.-L. Drechsler, H. Berger,
A. Erb, M. Lambacher, L. Patthey, V. Hinkov, and B. Keimer, Phys. Rev. B
76, 064519 (2007).50V. B. Zabolotnyy, S. V. Borisenko, A. A. Kordyuk, D. S. Inosov, A.
Koitzsch, J. Geck, J. Fink, M. Knupfer, B. B€uchner, S.-L. Drechsler, V.
Hinkov, B. Keimer, and L. Patthey, Phys. Rev. B 76, 024502 (2007).51A. A. Kordyuk, S. V. Borisenko, M. Knupfer, and J. Fink, Phys. Rev. B
67, 064504 (2003).52A. A. Kordyuk and S. V. Borisenko, Fiz. Nizk. Temp. 32, 401 (2006)
[Low Temp. Phys. 32, 298 (2006)].53T. Kondo, T. Takeuchi, T. Yokoya, S. Tsuda, S. Shin, and U. Mizutani,
J. Electron Spectrosc. Relat. Phenom. 137–140, 663 (2004).
54A. Kaminski, S. Rosenkranz, H. M. Fretwell, M. R. Norman, M.
Randeria, J. C. Campuzano, J.-M. Park, Z. Z. Li, and H. Raffy, Phys.
Rev. B 73, 174511 (2006).55A. Ino, C. Kim, M. Nakamura, T. Yoshida, T. Mizokawa, A. Fujimori,
Z.-X. Shen, T. Kakeshita, H. Eisaki, and S. Uchida, Phys. Rev. B 65,
094504 (2002).56A. A. Kordyuk, V. B. Zabolotnyy, D. V. Evtushinsky, D. S. Inosov, T. K.
Kim, B. B€uchner, and S. V. Borisenko, Eur. Phys. J. Spec. Top. 188, 153
(2010).57T. Valla, A. V. Fedorov, P. D. Johnson, and S. L. Hulbert, Phys. Rev.
Lett. 83, 2085 (1999).58A. A. Kordyuk, S. V. Borisenko, A. Koitzsch, J. Fink, M. Knupfer, and
H. Berger, Phys. Rev. B 71, 214513 (2005).59D. S. Inosov, J. Fink, A. A. Kordyuk, S. V. Borisenko, V. B. Zabolotnyy,
R. Schuster, M. Knupfer, B. B€uchner, R. Follath, H. A. D€urr, W.
Eberhardt, V. Hinkov, B. Keimer, and H. Berger, Phys. Rev. Lett. 99,
237002 (2007).60S. V. Borisenko, A. A. Kordyuk, T. K. Kim, A. Koitzsch, M. Knupfer, J.
Fink, M. S. Golden, M. Eschrig, H. Berger, and R. Follath, Phys. Rev.
Lett. 90, 207001 (2003).61T. K. Kim, A. A. Kordyuk, S. V. Borisenko, A. Koitzsch, M. Knupfer, H.
Berger, and J. Fink, Phys. Rev. Lett. 91, 167002 (2003).62A. Kaminski, M. Randeria, J. C. Campuzano, M. R. Norman, H. Fretwell,
J. Mesot, T. Sato, T. Takahashi, and K. Kadowaki, Phys. Rev. Lett. 86,
1070 (2001).63A. A. Kordyuk, S. V. Borisenko, A. Koitzsch, J. Fink, M. Knupfer, B.
B€uchner, H. Berger, G. Margaritondo, C. T. Lin, B. Keimer, S. Ono, and
Y. Ando, Phys. Rev. Lett. 92, 257006 (2004).64A. A. Kordyuk, S. V. Borisenko, V. B. Zabolotnyy, J. Geck, M. Knupfer,
J. Fink, B. B€uchner, C. T. Lin, B. Keimer, H. Berger, A. V. Pan, S.
Komiya, and Y. Ando, Phys. Rev. Lett. 97, 017002 (2006).65T. Dahm, V. Hinkov, S. V. Borisenko, A. A. Kordyuk, V. B. Zabolotnyy,
J. Fink, B. B€uchner, D. J. Scalapino, W. Hanke, and B. Keimer, Nat.
Phys. 5, 217 (2009).66M. Eschrig, Adv. Phys. 55, 47 (2006).67U. Chatterjee, D. K. Morr, M. R. Norman, M. Randeria, A. Kanigel, M.
Shi, E. Rossi, A. Kaminski, H. M. Fretwell, S. Rosenkranz, K. Kadowaki,
and J. C. Campuzano, Phys. Rev. B 75, 172504 (2007).68D. S. Inosov, S. V. Borisenko, I. Eremin, A. A. Kordyuk, V. B.
Zabolotnyy, J. Geck, A. Koitzsch, J. Fink, M. Knupfer, B. B€uchner, H.
Berger, and R. Follath, Phys. Rev. B 75, 172505 (2007).69A. A. Kordyuk, Fiz. Nizk. Temp. 41, 417 (2015) [Low Temp. Phys. 41,
319 (2015)].70P. Monthoux and D. Pines, Phys. Rev. B 49, 4261 (1994).71A. Abanov, A. V. Chubukov, M. Eschrig, M. R. Norman, and J.
Schmalian, Phys. Rev. Lett. 89, 177002 (2002).72T. A. Maier, A. Macridin, M. Jarrell, and D. J. Scalapino, Phys. Rev. B
76, 144516 (2007).73H.-Y. Kee, S. A. Kivelson, and G. Aeppli, Phys. Rev. Lett. 88, 257002
(2002).74A. S. Alexandrov, J. Phys.: Condens. Matter 19, 125216 (2007).75A. A. Kordyuk, Fiz. Nizk. Temp. 38, 1119 (2012) [Low Temp. Phys. 38,
888 (2012)].76A. A. Kordyuk, V. B. Zabolotnyy, D. V. Evtushinsky, A. N. Yaresko, B.
B€uchner, and S. V. Borisenko, J. Supercond. Novel Magn. 26, 2837 (2013).77O. K. Andersen and L. Boeri, Ann. Phys. 523, 8 (2011).78M. Sadovskii, E. Kuchinskii, and I. Nekrasov, in Fifth Moscow
International Symposium on Magnetism [J. Magn. Magn. Mater. 324,
3481 (2012)].79V. B. Zabolotnyy, D. S. Inosov, D. V. Evtushinsky, A. Koitzsch, A. A.
Kordyuk, G. L. Sun, J. T. Park, D. Haug, V. Hinkov, A. V. Boris, C. T.
Lin, M. Knupfer, A. N. Yaresko, B. B€uchner, A. Varykhalov, R. Follath,
and S. V. Borisenko, Nature 457, 569 (2009).80S. V. Borisenko, V. B. Zabolotnyy, D. V. Evtushinsky, T. K. Kim, I. V.
Morozov, A. N. Yaresko, A. A. Kordyuk, G. Behr, A. Vasiliev, R.
Follath, and B. B€uchner, Phys. Rev. Lett. 105, 067002 (2010).81A. A. Kordyuk, V. B. Zabolotnyy, D. V. Evtushinsky, T. K. Kim, I. V.
Morozov, M. L. Kulic, R. Follath, G. Behr, B. B€uchner, and S. V.
Borisenko, Phys. Rev. B 83, 134513 (2011).82J. Maletz, V. B. Zabolotnyy, D. V. Evtushinsky, S. Thirupathaiah, A. U.
B. Wolter, L. Harnagea, A. N. Yaresko, A. N. Vasiliev, D. A. Chareev,
A. E. B€ohmer, F. Hardy, T. Wolf, C. Meingast, E. D. L. Rienks, B.
B€uchner, and S. V. Borisenko, Phys. Rev. B 89, 220506 (2014).83L. Fanfarillo, J. Mansart, P. Toulemonde, H. Cercellier, P. Le Fevre, F.
Bertran, B. Valenzuela, L. Benfatto, and V. Brouet, Phys. Rev. B 94,
155138 (2016).
484 Low Temp. Phys. 44 (6), June 2018 A. A. Kordyuk
84D. V. Evtushinsky, A. N. Yaresko, V. B. Zabolotnyy, J. Maletz, T. K.
Kim, A. A. Kordyuk, M. S. Viazovska, M. Roslova, I. Morozov, R. Beck,
S. Aswartham, L. Harnagea, S. Wurmehl, H. Berger, V. A. Rogalev, V.
N. Strocov, T. Wolf, N. D. Zhigadlo, B. B€uchner, and S. V. Borisenko,
Phys. Rev. B 96, 060501 (2017).85M. Yi, D. H. Lu, J. G. Analytis, J.-H. Chu, S.-K. Mo, R.-H. He, R. G.
Moore, X. J. Zhou, G. F. Chen, J. L. Luo, N. L. Wang, Z. Hussain, D. J.
Singh, I. R. Fisher, and Z.-X. Shen, Phys. Rev. B 80, 024515 (2009).86V. Brouet, P.-H. Lin, Y. Texier, J. Bobroff, A. TalebIbrahimi, P. Le
Fevre, F. Bertran, M. Casula, P. Werner, S. Biermann, F. Rullier-
Albenque, A. Forget, and D. Colson, Phys. Rev. Lett. 110, 167002
(2013).87M. D. Watson, T. K. Kim, A. A. Haghighirad, N. R. Davies, A.
McCollam, A. Narayanan, S. F. Blake, Y. L. Chen, S. Ghannadzadeh, A.
J. Schofield, M. Hoesch, C. Meingast, T. Wolf, and A. I. Coldea, Phys.
Rev. B 91, 155106 (2015).88Y. V. Pustovit and A. A. Kordyuk, Fiz. Nizk. Temp. 42, 1268 (2016)
[Low Temp. Phys. 42, 995 (2016)].89F. Lochner, F. Ahn, T. Hickel, and I. Eremin, Phys. Rev. B 96, 094521
(2017).90H. Ding, P. Richard, K. Nakayama, K. Sugawara, T. Arakane, Y. Sekiba,
A. Takayama, S. Souma, T. Sato, T. Takahashi, Z. Wang, X. Dai, Z.
Fang, G. F. Chen, J. L. Luo, and N. L. Wang, Europhys. Lett. 83, 47001
(2008).91D. V. Evtushinsky, D. S. Inosov, V. B. Zabolotnyy, M. S. Viazovska, R.
Khasanov, A. Amato, H.-H. Klauss, H. Luetkens, Ch. Niedermayer, G. L.
Sun, V. Hinkov, C. T. Lin, A. Varykhalov, A. Koitzsch, M. Knupfer, B.
B€uchner, A. A. Kordyuk, and S. V. Borisenko, New J. Phys. 11, 055069
(2009).92S. V. Borisenko, V. B. Zabolotnyy, A. A. Kordyuk, D. V. Evtushinsky, T.
K. Kim, I. V. Morozov, R. Follath, and B. B€uchner, Symmetry 4, 251
(2012).93D. V. Evtushinsky, V. B. Zabolotnyy, T. K. Kim, A. A. Kordyuk, A. N.
Yaresko, J. Maletz, S. Aswartham, S. Wurmehl, A. V. Boris, D. L. Sun,
C. T. Lin, B. Shen, H. H. Wen, A. Varykhalov, R. Follath, B. B€uchner,
and S. V. Borisenko, Phys. Rev. B 89, 064514 (2014).94R. M. Fernandes, A. V. Chubukov, and J. Schmalian, Nat. Phys. 10, 97
(2014).95P. J. Hirschfeld, M. M. Korshunov, and I. I. Mazin, Rep. Prog. Phys. 74,
124508 (2011).96W.-G. Yin, C.-C. Lee, and W. Ku, Phys. Rev. Lett. 105, 107004 (2010).97S. Onari and H. Kontani, Phys. Rev. Lett. 109, 137001 (2012).98A. Chubukov, Annu. Rev. Condens. Matter Phys. 3, 57 (2012).99A. V. Chubukov, M. Khodas, and R. M. Fernandes, Phys. Rev. X 6,
041045 (2016).100F. Wang and D.-H. Lee, Science 332, 200 (2011).101J. J. Leem, F. T. Schmitt, R. G. Moore, S. Johnston, Y.-T. Cui, W. Li, M.
Yi, Z. K. Liu, M. Hashimoto, Y. Zhang, D. H. Lu, T. P. Devereaux, D.-H.
Lee, and Z.-X. Shen, Nature 515, 245 (2014).102D. V. Evtushinsky, A. A. Kordyuk, V. B. Zabolotnyy, D. S. Inosov, T. K.
Kim, B. B€uchner, H. Luo, Z. Wang, H.-H. Wen, G. Sun, C. Lin, and S.
V. Borisenko, J. Phys. Soc. Jpn. 80, 023710 (2011).103S. Aswartham, M. Abdel-Hafiez, D. Bombor, M. Kumar, A. U. B.
Wolter, C. Hess, D. V. Evtushinsky, V. B. Zabolotnyy, A. A. Kordyuk, T.
K. Kim, S. V. Borisenko, G. Behr, B. B€uchner, and S. Wurmehl, Phys.
Rev. B 85, 224520 (2012).104D. V. Evtushinsky, V. B. Zabolotnyy, L. Harnagea, A. N. Yaresko, S.
Thirupathaiah, A. A. Kordyuk, J. Maletz, S. Aswartham, S. Wurmehl, E.
Rienks, R. Follath, B. B€uchner, and S. V. Borisenko, Phys. Rev. B 87,
094501 (2013).105C. Liu, A. D. Palczewski, R. S. Dhaka, T. Kondo, R. M. Fernandes, E. D.
Mun, H. Hodovanets, A. N. Thaler, J. Schmalian, S. L. Bud’ko, P. C.
Canfield, and A. Kaminski, Phys. Rev. B 84, 020509 (2011).106C. He, Y. Zhang, B. P. Xie, X. F. Wang, L. X. Yang, B. Zhou, F. Chen,
M. Arita, K. Shimada, H. Namatame, M. Taniguchi, X. H. Chen, J. P.
Hu, and D. L. Feng, Phys. Rev. Lett. 105, 117002 (2010).107S. Thirupathaiah, D. V. Evtushinsky, J. Maletz, V. B. Zabolotnyy, A. A.
Kordyuk, T. K. Kim, S. Wurmehl, M. Roslova, I. Morozov, B. B€uchner,
and S. V. Borisenko, Phys. Rev. B 86, 214508 (2012).108D. Mou, S. Liu, X. Jia, J. He, Y. Peng, L. Zhao, L. Yu, G. Liu, S. He, X.
Dong, J. Zhang, H. Wang, C. Dong, M. Fang, X. Wang, Q. Peng, Z.
Wang, S. Zhang, F. Yang, Z. Xu, C. Chen, and X. J. Zhou, Phys. Rev.
Lett. 106, 107001 (2011).109Y. Zhang, L. X. Yang, M. Xu, Z. R. Ye, F. Chen, C. He, H. C. Xu, J.
Jiang, B. P. Xie, J. J. Ying, X. F. Wang, X. H. Chen, J. P. Hu, M.
Matsunami, S. Kimura, and D. L. Feng, Nat. Mater. 10, 273 (2011).
110J. Maletz, V. B. Zabolotnyy, D. V. Evtushinsky, A. N. Yaresko, A. A.
Kordyuk, Z. Shermadini, H. Luetkens, K. Sedlak, R. Khasanov, A.
Amato, A. Krzton-Maziopa, K. Conder, E. Pomjakushina, H.-H. Klauss,
E. D. L. Rienks, B. B€uchner, and S. V. Borisenko, Phys. Rev. B 88,
134501 (2013).111S. Thirupathaiah, T. St€urzer, V. B. Zabolotnyy, D. Johrendt, B. B€uchner,
and S. V. Borisenko, Phys. Rev. B 88, 140505 (2013).112A. Charnukha, S. Thirupathaiah, V. B. Zabolotnyy, B. B€uchner, N. D.
Zhigadlo, B. Batlogg, A. N. Yaresko, and S. V. Borisenko, Sci. Rep. 5,
10392 (2015).113A. Charnukha, D. V. Evtushinsky, C. E. Matt, N. Xu, M. Shi, B. B€uchner,
N. D. Zhigadlo, B. Batlogg, and S. V. Borisenko, Sci. Rep. 5, 18273
(2015).114S. Medvedev, T. M. McQueen, I. A. Troyan, T. Palasyuk, M. I. Eremets,
R. J. Cava, S. Naghavi, F. Casper, V. Ksenofontov, G. Wortmann, and C.
Felser, Nat. Mater. 8, 630 (2009).115J. Guo, S. Jin, G. Wang, S. Wang, K. Zhu, T. Zhou, M. He, and X. Chen,
Phys. Rev. B 82, 180520 (2010).116S. L. Skornyakov, V. I. Anisimov, D. Vollhardt, and I. Leonov, preprint
arXiv:1802.01850 (2018).117A. A. Kordyuk, S. V. Borisenko, V. B. Zabolotnyy, R. Schuster, D. S.
Inosov, D. V. Evtushinsky, A. I. Plyushchay, R. Follath, A. Varykhalov,
L. Patthey, and H. Berger, Phys. Rev. B 79, 020504 (2009).118M. Hashimoto, R.-H. He, K. Tanaka, J.-P. Testaud, W. Meevasana, R. G.
Moore, D. Lu, H. Yao, Y. Yoshida, H. Eisaki, T. P. Devereaux, Z.
Hussain, and Z.-X. Shen, Nat. Phys. 6, 414 (2010).119K. Rossnagel, J. Phys.: Condens. Matter 23, 213001 (2011).120D. Haug, V. Hinkov, Y. Sidis, P. Bourges, N. B. Christensen, A. Ivanov,
T. Keller, C. T. Lin, and B. Keimer, New J. Phys. 12, 105006 (2010).121M. Hashimoto, I. M. Vishik, R.-H. He, T. P. Devereaux, and Z.-X. Shen,
Nat. Phys. 10, 483 (2014).122I. S. Jacobs, J. W. Bray, H. R. Hart, Jr., L. V. Interrante, J. S. Kasper, G.
D. Watkins, D. E. Prober, and J. C. Bonner, Phys. Rev. B 14, 3036
(1976).123J. E. Hirsch, Phys. Rev. Lett. 53, 2327 (1984).124J. Spałek and A. Ole�s, Physica BþC 86–88, 375 (1977).125J. Spałek, Acta Phys. Pol. A 111, 409 (2007).126V. M. Loktev, Fiz. Nizk. Temp. 31, 645 (2005) [Low Temp. Phys. 31,
490 (2005)].127A. Koitzsch, S. V. Borisenko, A. A. Kordyuk, T. K. Kim, M. Knupfer, J.
Fink, M. S. Golden, W. Koops, H. Berger, B. Keimer, C. T. Lin, S. Ono,
Y. Ando, and R. Follath, Phys. Rev. B 69, 220505 (2004).128A. Mans, I. Santoso, Y. Huang, W. K. Siu, S. Tavaddod, V. Arpiainen,
M. Lindroos, H. Berger, V. N. Strocov, M. Shi, L. Patthey, and M. S.
Golden, Phys. Rev. Lett. 96, 107007 (2006).129L. Alff, Y. Krockenberger, B. Welter, M. Schonecke, R. Gross, D.
Manske, and M. Naito, Nature 422, 698 (2003).130T. Das, R. S. Markiewicz, and A. Bansil, Phys. Rev. B 81, 174504
(2010).131C. Weber, K. Haule, and G. Kotliar, Nat. Phys. 6, 574 (2010).132H. Matsui, K. Terashima, T. Sato, T. Takahashi, S.-C. Wang, H.-B.
Yang, H. Ding, T. Uefuji, and K. Yamada, Phys. Rev. Lett. 94, 047005
(2005).133S. R. Park, Y. S. Roh, Y. K. Yoon, C. S. Leem, J. H. Kim, B. J. Kim, H.
Koh, H. Eisaki, N. P. Armitage, and C. Kim, Phys. Rev. B 75, 060501
(2007).134A. Varlamov, V. Egorov, and A. Pantsulaya, Adv. Phys. 38, 469 (1989).135Y. Blanter, M. I. Kaganov, A. V. Pantsulaya, and A. A. Varlamov, Phys.
Rep. 245, 159 (1994).136V. I. Makarov, V. G. Bar’yakhtar, and V. V. Gann, Sov. Phys. JETP 40,
85 (1975).137E. A. Pashitskii and A. S. Shpigel, Ukr. J. Phys. 23, 669 (1978).138A. Perali, A. Bianconi, A. Lanzara, and N. L. Saini, Solid State Commun.
100, 181 (1996).139R. Caivano, M. Fratini, N. Poccia, A. Ricci, A. Puri, Z.-A. Ren, X.-L.
Dong, J. Yang, W. Lu, Z.-X. Zhao, L. Barba, and A. Bianconi,
Supercond. Sci. Technol. 22, 014004 (2009).140D. Innocenti, N. Poccia, A. Ricci, A. Valletta, S. Caprara, A. Perali, and
A. Bianconi, Phys. Rev. B 82, 184528 (2010).141D. Innocenti, S. Caprara, N. Poccia, A. Ricci, A. Valletta, and A.
Bianconi, Supercond. Sci. Technol. 24, 015012 (2011).142A. Bianconi, Nat. Phys. 9, 536 (2013).143A. Bianconi, N. Poccia, A. O. Sboychakov, A. L. Rakhmanov, and K. I.
Kugel, Supercond. Sci. Technol. 28, 024005 (2015).144M. V. Mazziotti, A. Valletta, G. Campi, D. Innocenti, A. Perali, and A.
Bianconi, Europhys. Lett. 118, 37003 (2017).
Low Temp. Phys. 44 (6), June 2018 A. A. Kordyuk 485
145A. A. Kalenyuk, A. Pagliero, E. A. Borodianskyi, A. A. Kordyuk, and V.
M. Krasnov, Phys. Rev. Lett. 120, 067001 (2018).146Y. S. Kushnirenko, A. A. Kordyuk, A. V. Fedorov, E. Haubold, T.
Wolf, B. B€uchner, and S. V. Borisenko, Phys. Rev. B 96, 100504
(2017).147A. I. Coldea, J. D. Fletcher, A. Carrington, J. G. Analytis, A. F. Bangura,
J.-H. Chu, A. S. Erickson, I. R. Fisher, N. E. Hussey, and R. D.
McDonald, Phys. Rev. Lett. 101, 216402 (2008).148L. Ortenzi, E. Cappelluti, L. Benfatto, and L. Pietronero, Phys. Rev. Lett.
103, 046404 (2009)
149L. Benfatto and E. Cappelluti, Phys. Rev. B 83, 104516 (2011).150L. C. Rhodes, M. D. Watson, A. A. Haghighirad, M. Eschrig, and T. K.
Kim, Phys. Rev. B 95, 195111 (2017).151Y. V. Pustovit, V. Bezguba, and A. A. Kordyuk, Metallofiz. Noveishie
Tekhnol. 39, 709 (2017).152Y. V. Pustovit et al., Metallofiz. Noveishie Tekhnol. 40, 138 (2018).
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